Communications in Mathematical Physics

, Volume 357, Issue 1, pp 295–317 | Cite as

A No-Go Theorem for the Continuum Limit of a Periodic Quantum Spin Chain

  • Vaughan F. R. Jones


We show that the Hilbert space formed from a block spin renormalization construction of a cyclic quantum spin chain (based on the Temperley–Lieb algebra) does not support a chiral conformal field theory whose Hamiltonian generates translation on the circle as a continuous limit of the rotations on the lattice.


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  1. 1.
    Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, New York (1982)MATHGoogle Scholar
  2. 2.
    Cannon J.W., Floyd W.J., Parry W.R.: Introductory notes on Richard Thompson’s groups. L’Enseign. Math. 42, 215–256 (1996)MathSciNetMATHGoogle Scholar
  3. 3.
    Cirac J.I., Verstraete F.: Renormalization and tensor product states in spin chains and lattices. J.Phys. A Math. Theor. 42(50), 504004 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dehornoy, P., Digne, F., Godelle, E., Krammer, D., Michel, J.: Foundations of Garside theory. arXiv:1309.0796
  5. 5.
    Dehornoy P.: The group of parenthesized braids. Adv. Math. 205, 354–409 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Evenbly, G., Vidal, G.: Tensor network renormalization. Phys. Rev. Lett. 115, 180405 (2015) arXiv:1412.0732
  7. 7.
    Ghosh, S.K., Jones, C.: Annular representation theory of rigid C*-tensor categories. J. Funct. Anal. 270, 1537–1584 (2016)Google Scholar
  8. 8.
    Graham J.J., Lehrer G.I.: The representation theory of affine Temperley–Lieb algebras. L’Enseign. Math. 44, 1–44 (1998)MathSciNetMATHGoogle Scholar
  9. 9.
    Jones, V.F.R.: Planar algebras I, preprint. arXiv:math/9909027
  10. 10.
    Jones V.F.R.: On knot invariants related to some statistical mechanical models. Pac. J. Math. 137, 311–334 (1989)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jones V.F.R.: The annular structure of subfactors, in “essays on geometry and related topics”. Monogr. Enseign. Math. 38, 401–463 (2001)Google Scholar
  12. 12.
    Jones, V.F.R.: In and around the origin of quantum groups. Prospects in math. Phys. Contemp. Math., 437 Am. Math. Soc. 101–126 (2007) arXiv:math.OA/0309199
  13. 13.
    Jones, V.F.R.: Some unitary representations of Thompson’s groups F and T (2014). arXiv:1412.7740
  14. 14.
    Jones V., Reznikoff S.: Hilbert Space representations of the annular Temperley–Lieb algebra. Pac. Math. J. 228, 219–250 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kauffman L.: State models and the Jones polynomial. Topology 26, 395–407 (1987)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Morrison S., Peters E., Snyder N.: Categories generated by a trivalent vertex. Sel. Math. New Ser. 23(2), 817–868 (2017)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pasquier V., Saleur H.: Common structures between finite systems and conformal field theories through quantum groups. Nucl. Phys. B 330(2), 523–556 (1990)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Penrose, R.: Applications of negative dimensional tensors. In: Welsh, D. (ed.) Applications of Combinatorial Mathematics, pp. 221–244. Academic Press, New York (1971)Google Scholar
  19. 19.
    Ren, Y.: From skein theory to presentations for Thompson group (2016). arXiv:1609.04077
  20. 20.
    Thomas, R.: An update on the four-color theorem. Not. AMS 45, 848–859 (1998)Google Scholar
  21. 21.
    Temperley H.N.V., Lieb E.H.: Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem. Proc. R. Soc. A 322, 251–280 (1971)ADSMathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Vanderbilt UniversityNashvilleUSA

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